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Two Examples of Borel Partially Ordered Sets with the Countable Chain Condition

1991
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Proceedings of the American Mathematical Society
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We define an open symmetric two-place relation on the reals such that the reals cannot be covered by countably many sets of related elements, but there is no uncountable set of unrelated elements. The poset & of finite sets of related elements satisfies the countable chain condition but it may fail to have the property K, i.e., a substantial irregularity can be injected in 5a . We construct two examples of Borel posets that satisfy the countable chain condition but violate certain natural

doi:10.2307/2048663
fatcat:hkpytd7m4bewjoy3bxrqbutu2m